GLOBAL STABILITY RESULTS FOR MODELS OF ...faculty.valpo.edu/dmaxin/IJB.pdfDANIEL MAXIN Department of...

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International Journal of Biomathematicsc© World Scientific Publishing Company

GLOBAL STABILITY RESULTS FOR MODELS OF

COMMENSALISM

PAUL GEORGESCU

Department of Mathematics, Technical University of Iasi,

Bd. Copou 11, 700506 Iasi, [email protected]

DANIEL MAXIN

Department of Mathematics and Computer Science, Valparaiso University,1900 Chapel Drive, Valparaiso, IN 46383, USA

[email protected]

HONG ZHANG

Department of Financial Mathematics, Jiangsu University,ZhenJiang, Jiangsu, 212013, P.R.China

[email protected]

Received (Day Mth. Year)

Revised (Day Mth. Year)

Abstract. We analyze the global stability of the coexisting equilibria for several models

of commensalism, first by devising a procedure to modify several Lyapunov functionals

which were introduced earlier for corresponding models of mutualism, further confirmingtheir usefulness. It is seen that commensalism promotes global stability, in connectionwith higher order self-limiting terms which prevent unboundedness. We then use the

theory of asymptotically autonomous systems to prove global stability results for mod-els of commensalism which are subject to Allee effects, finding that commensalisms of

appropriate strength can overcome the influence of strong Allee effects.

Keywords: Commensalism; global stability; Lyapunov functional; asymptotically au-tonomous systems.

1. Introduction

Depending on the beneficial (+), detrimental (–) or neutral (0) effects of two species

on each other’s ability to survive, grow or reproduce, interspecies interactions range

from mutualism (++) to competition (– –) (Holland and Bronstein [13]). Among

these possible types, the facilitative ones, in which benefits such as higher growth

rates, higher reproductive outputs or sheltering from external risk factors are en-

joyed by one or both species, are usually categorized as mutualism and commen-

salism. Mutualism (++) is therefore a specific form of facilitation in which the

1


2 P. Georgescu, D. Maxin & H. Zhang

association is beneficial for both species. Commensalism (+0), on the other hand,

describes a situation in which one species benefits from the association, while the

other is left unaffected.

An example of commensalism is the interaction between small plants called epi-

phytes and the tree on which they grow and from which they derive structural

support. Since epiphytes extract nutrients from the atmosphere, not from the tree,

and also do not harm or otherwise interfere with its growth in any way, there is

little data or reason to support any harm or benefit for the tree. Chemical commen-

salistic associations occur between bacterial species, one bacterium metabolizing a

chemical useless to the second bacterium and releasing residual metabolites which

are useful as a source of energy for the second bacterium, which is the sole ben-

eficiary of this association. Such interaction occurs between Acetobacter oxydans,

which produces fructose by oxidising mannitol and Saccharomyces carlsbergensis,

which can metabolise fructose, but cannot metabolise mannitol (Hogan [12]).

While a much higher attention has been devoted to antagonistic interactions

(competition and predation) than to the facilitative ones, commensalism has re-

ceived even less attention than mutualism. A common criticism to the concept of

commensalism is that while positive or negative effects are usually easily noticeable,

it is rather difficult to establish that a species is truly not affected by the interaction

with the other, being argued that interactions are more likely to be asymmetric,

one species exhibiting a strong positive response, while the other exhibits a weak

positive or negative response to the first (McCoy et al. [18]).

It has long been recognized that, due to a variety of biological reasons, a positive

correlation between the size of a population and its per capita growth rate, known

as the Allee effect, may exist at small densities. Two of the most common causes are

the benefits extracted from group defense or cooperation and an increased chance

of reproductive success due to easier mate finding (Courchamp et al. [3], Terry [19],

Guo et al. [9]).

By the nature of the density dependence, the Allee effects can be classified as

weak or strong. If the population exhibits a critical population size below of which

extinction is guaranteed, then the Allee effect is called strong, while if there is no

such critical threshold, then the Allee effect is called weak.

For instance, the classical logistic model

x′ = rx(

1− x

K

),

for which the per capita growth rate is f(x) = r(1− x

K

), decreasing for all popu-

lation densities, exhibits no Allee effects. On the contrary, the model

x′ = rx( xA− 1)(

1− x

K

),

for which the per capita growth rate is f(x) = r(xA − 1

) (1− x

K

), negative for

x ∈ (0, A), but with derivative f ′(x) = − 2xAK +

(1A + 1

K

), positive for small x,


Global stability results for models of commensalism 3

exhibits a strong Allee effect with threshold equal to A. Finally, the model

x′ = rxp(

1− x

K

), p > 1,

for which the per capita growth rate is f(x) = rxp−1(1− x

K

), positive for small x,

with derivative f ′(x) = rxp−2(p− 1− pxK ), also positive for small x, exhibits a weak

Allee effect.

2. Previous work on mutualistic models and motivation

In Vargas-De-Leon [22], the global stability of the positive equilibria for two-species

mutualisms has been investigated by means of Lyapunov’s second method. The

models of interest in [22] are

dx1dt

= r1x1

[(1− e1

r1

)− x1K1

]+r1b12K1

x1x2 (2.1)

dx2dt

= r2x2

[(1− e2

r2

)− x2K2

]+r2b21K2

x1x2,

introduced by Vandermeer and Boucher in [21] and

dx1dt

= (r1 − e1)x1 −r1x

21

K1 + b12x2(2.2)

dx2dt

= (r2 − e2)x2 −r2x

22

K2 + b21x1,

introduced by Wolin and Lawlor in [25]. In the above models, both representing

facultative mutualisms, ri represents the intrinsic birth rate of species xi, while

Ki and ei are the carrying capacity of the environment and the harvesting effort,

respectively, with regard to the same species xi, i = 1, 2. Also, b12 and b21 are

strictly positive constants quantifying the mutualistic support the species give to

each other. Both models were initially introduced without accounting for the effects

of harvesting. Also, if one species is missing, the other behaves in the same way in

both models, namely in a logistic fashion.

An abstract model of a mutualistic interaction, in the form

dx1dt

= a1(x1) + f1(x1)g1(x2) (2.3)

dx2dt

= a2(x2) + f2(x2)g2(x1),

has been studied by Georgescu and Zhang in [6], the real continuous functions a1,

a2, f1, f2, g1, g2 being assumed to satisfy several combinations of monotonicity

properties and sign conditions. The existence of a coexisting equilibrium E∗ =

(x∗1, x∗2) has been a priori assumed in [6], global stability properties for E∗ being

then obtained by means of Lyapunov’s second method. In [6], use has been made



of the following functionals

V1(x1, x2) =

∫ x1

x∗1

g2(θ)− g2(x∗1)

f1(θ)dθ +

∫ x2

x∗2

g1(θ)− g1(x∗2)

f2(θ)dθ, (2.4)

V2(x1, x2) =

∫ x1

x∗1

(1− g2(θ)

g2(x∗1)

)1

a1(θ)dθ +

∫ x2

x∗2

(1− g1(θ)

g1(x∗2)

)1

a2(θ)dθ,

V3(x1, x2) =

∫ x1

x∗1

(1− g2(x∗1)

g2(θ)

)1

f1(θ)dθ

+

[∫ x2

x∗2

(1− g1(x∗2)

g1(θ)

)1

f2(θ)dθ

]g1(x∗2)

g2(x∗1).

However, a number of assumptions in [6], despite of having a general nature, are of

a rather involved form, their applicability being subject of further investigation. To

further establish the usefulness of the functionals V1, V2, V3, Georgescu et. al. have

used them in [7] to prove the global stability of the positive equilibria for versions

of (2.1) and (2.2) featuring a Richards growth term in place of the logistic one, in

the form

dx1dt

= r1x1

[A1 −

(x1K1

)p]+r1b12K1

x1x2 (2.5)

dx2dt

= r2x2

[A2 −

(x2K2

)p]+r2b21K2

x1x2,

and respectively

dx1dt

= r1x1A1 −r1x

p+11

Kp1 + b12x2

(2.6)

dx2dt

= r2x2A2 −r2x

p+12

Kp2 + b21x1

.

In the above models, A1 = 1 − e1r1

, A2 = 1 − e2r2

and p ≥ 1, being assumed that

0 ≤ e1 < r1, 0 ≤ e2 < r2 and that b12, b21 > 0, K1,K2 > 0 and being also observed

that for p = 1 the models (2.5) and (2.6) reduce to (2.1) and (2.2), respectively.

The global stability of the positive equilibrium of the following mutualistic model

with restricted growth rates

dx1dt

= r1x1

(1− x1

K1

)+ c1x1(1− e−α2x2) (2.7)

dx2dt

= r2x2

(1− x2

K2

)+ c2x2(1− e−α1x1),

proposed by Graves et al. in [8] has also been established in [7] by using Lyapunov’s

second method and the functional V3.

One would think that the stability of the coexisting equilibria for two-species

models of commensalism would follow immediately from the corresponding results

for models of mutualism, when these results are available. After all, commensalism



can be thought as mutualism in which one of the two interspecies interaction terms

is zero, so at a glance everything should be simpler.

However, this is not actually the case. Even a cursory look at the specific forms

of the functionals V1, V2, V3 given in (2.4) (or of the functionals in [22], which are

particular cases) shows that they cannot deal directly with the situation in which

one or more of the functions f1, f2, g1, g2 are null, as either a denominator or one

of the integrals are null. By replacing the problematic integral (null or with null

denominator) with a logarithmic term, Vargas-De-Leon and Gomez-Alcaraz [23]

have obtained global stability results for the positive equilibria of the models

dx1dt

= r1x1

(1− x1

K1

)(2.8)

dx2dt

= r2x2

(1− x2

K2

)+r2b21K2

x1x2,

and

dx1dt

= r1x1

(1− x1

K1

)(2.9)

dx2dt

= r2x2

(1− x2

K2 + b21x1

),

which are the direct commensalistic counterparts of (2.1) and (2.2). Although the

use of logarithmic functionals to establish the stability of two-species models has

a tradition which goes back to Volterra [24], one would, of course, think of a mo-

tivation of their use and what to use in place of a logarithm for different looking

two-species models. The Lyapunov functional V1 has been introduced (up to a sign

change) by Harrison in [11] to discuss the stability of a predator-prey interaction.

Related functionals have been systematically employed by Korobeinikov [14,15] to

establish the stability of equilibria for general disease propagation models with

abstract nonlinear incidence. See also Georgescu and Hsieh [5], where the global

dynamics of a SEIV model with nonlinear incidence of infection and removal has

been studied using the same approach, employing a functional which is formally

related to V1 and McCluskey [17], who discussed the stability of a SEIR model with

varying infectivity and infinite delay.

On the same line of thought, we continue here the investigation started in [7]

and obtain the global stability of the positive equilibria for the models

dx1dt

= r1x1

[1−

(x1K1

)p](2.10)

dx2dt

= r2x2

[1−

(x2K2

)p]+r2b21K2

x1x2,



and respectively

dx1dt

= r1x1

[1−

(x1K1

)p](2.11)

dx2dt

= r2x2 −r2x

p+12

Kp2 + b21x1

,

that is, for the commensalistic counterparts of (2.5) and (2.6), respectively, without

harvesting (so that e1 = e2 = 0 and consequently A1 = A2 = 1), assuming that

p ≥ 1. We also establish the global stability of the positive equilibrium of the

following commensalistic model with restricted growth rate

dx1dt

= r1x1

(1− x1

K1

)(2.12)

dx2dt

= r2x2

(1− x2

K2

)+ c2x2(1− e−α1x1),

that is, the commensalistic correspondent of (2.7).

Another motivation for studying commensalistic models separately is related to

their criticism mentioned earlier. Even if commensalism is in reality a special form

of strictly positive (but asymmetric) two-way interaction between species, a math-

ematical continuity argument can still use the commensalism stability results we

derive in this paper as a good approximation. In other words, one can reasonably ex-

pect that a result for a commensalism model of type (2.3) with g1(x2) = 0 is a good

approximation for a mutualistic model with g1(x2) close to zero in a biologically

suitable interval for x2.

At the same time, it may not be possible to simply use pre-existing theorems

proved for mutualistic models by simply adjusting the parameters to force one of

these functions to be close to zero. This may be due to specific biological assump-

tions in situations where one of the species has a negligible effect on the other. Some

arguments that criticize commensalism models mention the fact that at higher den-

sities the apparently neutral species may have a non-zero effect on the other. For

example, Epiphytes, which are non-parasitic plants that grow attached to trees, may

impede the photosynthesis of the host plant at very high densities (negative effect)

or they may protect the bark of the host (positive effect) (Benzing [1]).

Therefore, it may not be possible to adjust, for example, a mutualistic model

already studied to simmulate a “near commensalism” situation. For example, in the

mutualistic model with restricted growth rate (2.7), if the neutral species is given

by x2 then one cannot simply consider g1(x2) = c1(1− e−α2x2) by using a c1 close

to zero to simmulate a commensalism interaction since g1 is a concave downward

function which indicates that the impact rate of x2 actually decreases with higher

values of x2 which is the opposite of what may happen in reality. Same remark can

be made in the case of (2.5) and (2.6). In the first, g1 would be a linear function

which suggests that the impact rate of neutral species grows at a constant rate



irrespective of its density; in the latter, g1 is actually a decreasing function in x2,

again contrary to what may happen in a nearly commensalistic interaction.

3. Algebraic tools

We now introduce several inequalities which will be useful when evaluating the

derivatives of Lyapunov functionals involved in the proofs of the stability results

for the coexisting equilibria of (2.10), (2.11) and (2.12). Their asymmetric nature

is due to the asymmetric nature of the commensalistic models to which they are to

be applied.

Lemma 3.1. The following inequalities hold.

(1) If u > 0 and p ≥ 0, then

(1− 1

u

)(1− up) ≤ 0, (3.1)

with equality if p = 0 and all u > 0 or if p > 1 and u = 1.

(2) If u, v > 0 and p ≥ 1, then

(1− 1

u

)(1− up) +

(1− 1

v

)(u− vp) ≤ 0, (3.2)

with equality if and only if u = v = 1.

(3) If u, v > 0 and p ≥ 1, then

1

up−1

(1− 1

u

)(1− up) +

1

vp−1

(1− 1

v

)(u− vp) ≤ 0, (3.3)


(4) If u, v > 0 and p ≥ 1, then

(u− 1) (1− up) + (v − 1) (u− vp) ≤ 0, (3.4)


Proof. The first inequality is obvious, and so is its equality case. For the second

one, let us denote

E1 =

(1− 1

u

)(1− up) +

(1− 1

v

)(u− vp)



and observe that

E1 =

(1− 1

u

)(u− up) +

(1− 1

u

)(1− u) +

(1− 1

v

)(v − vp)

+

(1− 1

v

)(u− v)

= u

(1− 1

u

)(1− up−1

)+ v

(1− 1

v

)(1− vp−1

)+ 3− 1

u− v − u

v

= E11 + E12 + E13.

While E11 ≤ 0 and E12 ≤ 0 by (3.1), it is also seen that E13 ≤ 0 by AM − GMinequality (i.e. the inequality between the arithmetic mean and the geometric mean

for two positive numbers). Since E13 = 0 if and only if u = v = 1, case which leads

also to E11 = E12 = 0, it is seen that the equality case for (3.2) is u = v = 1. To

prove (3.3), let us denote

E2 =1

up−1

(1− 1

u

)(1− up) +

1

vp−1

(1− 1

v

)(u− vp)

and observe that

E2 =1

up−1

(1− 1

u

)(1− up−1

)+

1

up−1

(1− 1

u

)(up−1 − up

)+

1

vp−1

(1− 1

v

)(u− uvp−1

)+

1

vp−1

(1− 1

v

)(uvp−1 − vp

)=

1

up−1

(1− 1

u

)(1− up−1

)+

(1− 1

u

)(1− u)

+u

vp−1

(1− 1

v

)(1− vp−1

)+

(1− 1

v

)(u− v)

= E21 + E22 + E23 + E24.

By (3.1), it is seen that E21 ≤ 0 and E23 ≤ 0. Also,

E22 + E24 = 3− 1

u− v − u

v≤ 0,

as seen above, by AM−GM inequality, from which we also obtain that the equality

case for (3.3) is u = v = 1. To deduce (3.4), let us denote

E3 = (u− 1) (1− up) + (v − 1) (u− vp) ,



and observe that

E3 = (u− 1)(1− u) + (u− 1) (u− up) + (v − 1)(u− v) + (v − 1) (v − vp)

= (−u2 − v2 + u+ v − 1 + uv) + u2(

1− 1

u

)(1− up−1

)+ v2

(1− 1

v

)(1− vp−1

)= −1

2

[(u− 1)2 + (v − 1)2 + (u− v)2

]+ u2

(1− 1

u

)(1− up−1

)+ v2

(1− 1

v

)(1− vp−1

)= E31 + E32 + E33.

It is obvious that E31 ≤ 0, while E32 ≤ 0 and E33 ≤ 0 by (3.1). Noting that E31 = 0

if and only if u = v = 1, the equality case for (3.4) is immediate.

4. Commensalistic models with Richards growth

We start with an analysis of the models (2.10) and (2.11). It is easily seen that

(0,∞)× (0,∞) is an invariant region for both of them. Also, the models (2.10) and

(2.11) admit the coexisting equilibria E∗1 and E∗2, respectively, with

E∗1 = (K1,(Kp

2 + b21K1Kp−12

) 1p

), E∗2 = (K1, (Kp2 + b21K1)

1p ). (4.1)

We shall now proceed with a stability analysis of the coexisting equilibria E∗1 and

E∗2. Let us start with E∗1 and denote E∗1 = (x∗1, x∗2). It is seen that the components

x∗1 and x∗2, given in (4.1), also satisfy the following equilibrium relation

1 +b21K2

x∗1 =

(x∗2K2

)p. (4.2)

For the proof of our stability results, we shall employ Lyapunov’s second method.

We now give a motivation regarding the specific form of the functional we are going

to employ. It has been observed in Georgescu et al. [7] that the following particular

form of V3

V3(x1, x2) =

∫ x1

x∗1

(1− x∗1

θ

)1

θdθ +

r1b12K2x∗2

r2b21K1x∗1

[∫ x2

x∗2

(1− x∗2

θ

)1

θdθ

]can be used to prove the global stability of the positive equilibrium of (2.5), that

is, of a mutualistic version of (2.10) with harvesting, for the following choice of f1,

f2, f3 and f4 in the abstract framework (2.3)

a1(x1) = r1x1

[A1 −

(x1K1

)p], a2(x2) = r2x2

[A2 −

(x2K2

)p],

f1(x1) = x1, g1(x2) =r1b12x2K1

, f2(x2) = x2, g2(x1) =r2b21x1K2

.



To construct our Lyapunov functional using V3 as a template, we note that, for our

model (2.10), b12 = 0 and g1 is null. Although the abstract form of V3 given in

(2.4) involves the fractiong1(x

∗2)

g1(θ)in the second integral (which is the troublesome

one), the integrand can be left unchanged in the concrete form of V3 given above if

one thinks that the simplification of b12 (which is now 0) occurs beforehand. The

coefficient of the second integral is also 0, but there is no possibility of having a

Lyapunov functional consisting of the first integral alone, so we may need a positive

constant defined ad hoc in place of b12. This heuristic argument leads us to using

the Lyapunov functional

U3(x1, x2) =

∫ x1

x∗1

(1− x∗1

θ

)1

θdθ +

r1αK2x∗2

r2b21K1x∗1

[∫ x2

x∗2

(1− x∗2

θ

)1

θdθ

],

(we have chosen the subscript to stress the originating template functional, although

there are no U1 and U2 defined as of yet), where α > 0 will be determined later on by

matching coefficients. It is of no surprise to remark that the coexisting equilibrium

E∗1 preserves the global asymptotic stability property which has been proved in [7]

for the mutualistic counterpart of (2.10).

Theorem 4.1. The coexisting equilibrium E∗1 is globally asymptotically stable in

(0,∞)× (0,∞).

Proof. First, it is seen that U3 increases whenever any of |x1 − x∗1| and |x2 − x∗2|increases and that U3(x1, x2) ≥ 0, with equality if and only if x1 = x∗1 and x2 = x∗2.

Also, the level sets of U3 do not have limit points on the boundary of (0,∞)×(0,∞),

since U3(x1, x2) tends to ∞ if either x1 or x2 tends to 0 or to ∞.

We now evaluate the derivative of U3 along the solutions of (2.10). One sees

that

·U3 =

(1− x∗1

x1

)1

x1

dx1dt

+r1αK2x

∗2

r2b21K1x∗1

(1− x∗2

x2

)1

x2

dx2dt

(4.3)

= r1

(1− x∗1

x1

)[1−

(x1K1

)p]+r1αK2x

∗2

b21K1x∗1

(1− x∗2

x2

)[1−

(x2K2

)p+b21x1K2

].

By the equilibrium condition (4.2), one sees that(1 +

b21K2

x∗1

)(K2

x∗2

)p= 1,

and consequently, by substituting this relation into the right-hand side of (4.3) and



recalling from (4.1) that x∗1 = K1, we obtain

·U3 = r1

(1− x∗1

x1

)[1−

(x1x∗1

)p]+r1αK2x

∗2

b21(x∗1)2

(1− x∗2

x2

)[1−

(x2x∗2

)p(1 +

b21K2

x∗1

)+b21x1K2

]= r1

(1− x∗1

x1

)[1−

(x1x∗1

)p]+r1αK2x

∗2

b21(x∗1)2

(1− x∗2

x2

)[1−

(x2x∗2

)p]+

(1− x∗2

x2

)r1αx

∗2

x∗1

[x1x∗1−(x2x∗2

)p]= T1 + T2 + T3. (4.4)

By inequality (3.1) of Lemma 3.1, we observe that T2 ≤ 0. Let us now choose

α =x∗1

x∗2, so that

αx∗2

x∗1

= 1. Then, by inequality (3.2) of Lemma 3.1 with u = x1

x∗1

and

v = x2

x∗2, we obtain that T1+T3 ≤ 0, with equality if and only if x1 = x∗1 and x2 = x∗2.

The use of LaSalle’s invariance theorem completes the proof of Theorem 4.1.

We now turn our attention to the stability of E∗2. Let us denote again E∗2 =

(x∗1, x∗2), for an easier construction of the functional through similarity with the

templates given in (2.4), since there is no danger of confusion with the coordinates

of E∗1. It is seen that x∗1 and x∗2 satisfy the following equilibrium relation

Kp2 + b21x

∗1

(x∗2)p= 1. (4.5)

It has been observed in [7] that the following particular form of V1

V1(x1, x2) =A2b21

(x∗2)pr1

∫ x1

x∗1

θ − x∗1(Kp

2 + b21θ)θp+1dθ +

A1b12(x∗1)pr2

∫ x2

x∗2

θ − x∗2(Kp

1 + b12θ)θp+1dθ

can be used to prove the global stability of the positive equilibrium of (2.6), that

is, of the mutualistic version of (2.11). Since one could consider again that b12 = 0,

by a heuristic argument similar to the one displayed above and noting that now

A1 = A2 = 1, one could use the functional

U1(x1, x2) =b21

(x∗2)pr1

∫ x1

x∗1

θ − x∗1(Kp

2 + b21θ)θp+1dθ +

α

(x∗1)pr2

∫ x2

x∗2

θ − x∗2Kp

1θp+1

dθ,

where α > 0 shall be determined later on by matching coefficients.


(0,∞)× (0,∞).


Also, the level sets of U1 do not have limit points on the boundary of (0,∞)×(0,∞)




We now evaluate the derivative of U1 along the solutions of (2.10). One sees

that·U1 =

b21(x∗2)pr1

x1 − x∗1(Kp

2 + b21x1)xp+11

dx1dt

+α

(x∗1)pr2

x2 − x∗2Kp

1xp+12

dx2dt

(4.6)

=b21

(x∗2)px1 − x∗1

(Kp2 + b21x1)xp1

[1−

(x1x∗1

)p]+

α

Kp1 (x∗1)p

x2 − x∗2(Kp

2 + b21x1)xp2(Kp

2 + b21x1 − xp2) .

By substituting (4.5) into the right-hand side of (4.6), it is then seen that

·U1 =

b21(x∗2)p

x1 − x∗1(Kp

2 + b21x1)xp1

[1−

(x1x∗1

)p]+

α

Kp1 (x∗1)p

x2 − x∗2(Kp

2 + b21x1)xp2

(Kp

2 + b21x1 −Kp

2 + b21x∗1

(x∗2)pxp2

)=

b21(x∗2)p

x1 − x∗1(Kp

2 + b21x1)xp1

[1−

(x1x∗1

)p]+

α

Kp1 (x∗1)p

x2 − x∗2(Kp

2 + b21x1)xp2Kp

2

[1−

(x2x∗2

)p]+

α

Kp1 (x∗1)p

x2 − x∗2(Kp

2 + b21x1)xp2b21x

∗1

[x1x∗1−(x2x∗2

)p]= T1 + T2 + T3.

We now evaluate the signs of T2 and of T1+T3 with the help of Lemma 3.1, choosing

in the process the right value for α. We observe that

T2 =α

Kp1 (x∗1)p

Kp2

(Kp2 + b21x1)xp−12

(1− x∗2

x2

)[1−

(x2x∗2

)p]≤ 0,

by inequality (3.1) of Lemma 3.1. Also,

T1 + T3 =b21

(Kp2 + b21x1) (x∗2)p(x∗1)p−1

{(x∗1x1

)p−1(1− x∗1

x1

)[1−

(x1x∗1

)p]

+αx∗2

(x∗1)p

(1− x∗2

x2

)(x∗2x2

)p−1 [x1x∗1−(x2x∗2

)p]}.

Let us now choose α =(x∗

1)p

x∗2

, so thatαx∗

2

(x∗1)

p = 1. Then

T1 + T3 =b21

(Kp2 + b21x1) (x∗2)p(x∗1)p−1

{(x∗1x1

)p−1(1− x∗1

x1

)[1−

(x1x∗1

)p]

+

(x∗2x2

)p−1(1− x∗2

x2

)[x1x∗1−(x2x∗2

)p]}≤ 0,



by inequality(3.3) of Lemma 3.1 with u = x1

x∗1

and v = x2

x∗2. It is also to be noted

that T1 +T3 = 0 if and only if x1 = x∗1 and x2 = x∗2. The use of LaSalle’s invariance

theorem completes the proof.

Let us now provide an alternate proof of Theorem 4.2, this time using as a

template the remaining mutualistic Lyapunov functional, V2. It has been observed

in [7] that the following particular form of V2,

V2(x1, x2) =b21r1A1

∫ x1

x∗1

θ − x∗1(Kp

2 + b21θ)θdθ +

b12r2A2

∫ x2

x∗2

θ − x∗2(Kp

1 + b12θ)θdθ.

is of use to prove the global stability of the positive equilibrium of (2.6), that is, of

the mutualistic version of (2.11). Motivated by this specific form of V2 and keeping

in mind that now A1 = A2 = 1, we shall attempt to use the following Lyapunov

functional

U2(x1, x2) =b21r1

∫ x1

x∗1

θ − x∗1(Kp

2 + b21θ)θdθ +

α

r2

∫ x2

x∗2

θ − x∗2Kp

1θdθ,

where α > 0 will be determined, as usual, by coefficient matching. We now evaluate

the derivative of U2 along the solutions of (2.11). It is seen that

·U2 =

b21r1

x1 − x∗1(Kp

2 + b21x1)x1

dx1dt

+α

r2

x2 − x∗2Kp

1x2

dx2dt

=b21

Kp2 + b21x1

(x1 − x∗1)

[1−

(x1x∗1

)]p+

α

Kp1

(x2 − x∗2)1

Kp2 + b21x1

(Kp2 + b21x1 − xp2) .

Using again the equilibrium relation (4.5), we observe that

·U2 =

b21Kp

2 + b21x1(x1 − x∗1)

[1−

(x1x∗1

)p]+

α

Kp1

x2 − x∗2Kp

2 + b21x1Kp

2

[1−

(x2x∗2

)p]+

α

Kp1

x2 − x∗2Kp

2 + b21x1b21x

∗1

[x1x∗1−(x2x∗2

)p]= T1 + T2 + T3.

We observe that

T2 =α

Kp1x2

Kp2

Kp2 + b21x1

(1− x∗2

x2

)[1−

(x2x∗2

)p]≤ 0,

by inequality (3.1) of Lemma 3.1. Also,

T1 + T3 =b21x

∗1

Kp2 + b21x1

{(x1x∗1− 1

)[1−

(x1x∗1

)p]+αx∗2Kp

1

(x2x∗2− 1

)[x1x∗1−(x2x∗2

)p]}.



Choosing now α =Kp

1

x∗2

, so thatαx∗

2

Kp1

= 1, it follows by inequality (3.4) of Lemma 3.1

that T1 + T3 ≤ 0, with equality if and only if x1 = x∗1 and x2 = x∗2. The use of

LaSalle’s invariance principle finishes this alternate proof.

5. A commensalistic model with restricted growth rates

We now discuss the stability properties of the model (2.12). Again, it is easily seen

that (0,∞)×(0,∞) is an invariant region for (2.12). Also, (2.12) admits the positive

equilibrium E∗3 = (x∗1, x∗2), given by

E∗3 = (K1,K2(1 +c2r2

(1− e−α1K1))). (5.1)

Note that x∗1 and x∗2 also satisfy the following equilibrium relation

1 =x∗2K2− c2r2

(1− e−α1x

∗1

). (5.2)

It has been observed in [7] that the following particular form of V3

V3(x1, x2) =

∫ x1

x∗1

e−α1x∗1 − e−α1θ

1− e−α1θ

1

c1θdθ

+

(∫ x2

x∗2


1− e−α2θ

1

c2θdθ

)1− e−α2x

∗2

1− e−α1x∗1.

is of use to prove the global stability of the positive equilibrium of (2.7), that is, of

the mutualistic version of (2.12).

To define the suitable Lyapunov functional for our model of commensalism, the

problem is twofold, since now two coefficients, c1 and α2, are null. Among these, c1is not paired (i.e., it does not appear both at the denominator and the numerator of

a fraction). Noting that a Lyapunov functional remains a Lyapunov functional after

multiplication with a nonzero constant and taking a few formal limits for α2 → 0

inside the second integral (again, the problematic one), we arrive at the following

tentative Lyapunov functional

U3(x1, x2) = c1

∫ x1

x∗1


1− e−α1θ

1

θdθ +

(∫ x2

x∗2

(1− x∗2

θ

)1

θdθ

)α

1− e−α1x∗1,

where α > 0 will be determined later on by the usual procedure.


(0,∞)× (0,∞).


Also, the level sets of U3 do not have limit points on the boundary of (0,∞)×(0,∞)




We now evaluate the derivative of U3 along the solutions of (2.12). It is seen

that·

U3 = c2e−α1x

∗1 − e−α1x1

1− e−α1x1

1

x1

dx1dt

+

(1− x∗2

x2

)1

x2

α

1− e−α1x∗1

dx2dt

= c2e−α1x

∗1 − e−α1x1

1− e−α1x1r1

(1− x1

K1

)+

(1− x∗2

x2

)α

1− e−α1x∗1r2

[1− x2

K2+c2r2

(1− e−α1x1)

]= T1 + T2.

Evaluating the second term T2, we observe that

T2 =

(1− x∗2

x2

)αr2

1− e−α1x∗1

[x∗2 − x2x∗2

+x2x∗2− x2K2

+c2r2

(1− e−α1x1)

]=

(1− x∗2

x2

)αr2

1− e−α1x∗1

(1− x2

x∗2

)+

(1− x∗2

x2

)αr2

1− e−α1x∗1

[x2x∗2− x2K2

+c2r2

(1− e−α1x1)

]= T21 + T22.

It is seen that T21 ≤ 0, with equality if and only if x2 = x∗2, by inequality (3.1) of

Lemma 3.1 or by AM −GM inequality. Also,

T22 =

(1− x∗2

x2

)αc2

1− e−α1x∗1

[r2c2

(x2x∗2− x2K2

)+ (1− e−α1x1)

](5.3)

=

(1− x∗2

x2

)αc2

1− e−α1x∗1

[r2c2

x2x∗2

(1− x∗2

K2

)+ (1− e−α1x1)

].

By the equilibrium relations (5.2) and (5.3), it is seen that

T22 =

(1− x∗2

x2

)αc2

1− e−α1x∗1

[−x2x∗2

(1− e−α1x∗1 ) + (1− e−α1x1)

]=

(1− x∗2

x2

)αc2

(−x2x∗2

+1− e−α1x1

1− e−α1x∗1

)Let us now define

Φ : (0,∞)→ (0,∞), Φ(x) =1− e−α1x

x.

Noting that

Φ′(x) = −e−α1x

x2(eα1x − (1 + α1x)) < 0,

it follows that Φ is strictly decreasing on (0,∞). With this notation,

T22 =

(1− x∗2

x2

)αc2

(−x2x∗2

+Φ(x1)x1Φ(x∗1)x∗1

).



Recalling that, by (5.1), x∗1 = K1, we obtain

T1 = c2

(1− 1− e−α1x

∗1

1− e−α1x1

)r1

(1− x1

K1

)= c2

(1− Φ(x∗1)x∗1

Φ(x1)x1

)r1

(1− x1

x∗1

).

This leads to

T1 + T22 = c2

(1− Φ(x∗1)x∗1

Φ(x1)x1

)r1

(1− x1

x∗1

)+

(1− x∗2

x2

)αc2

(−x2x∗2

+Φ(x1)x1Φ(x∗1)x∗1

)= c2r1

(1− x1

x∗1− Φ(x∗1)x∗1

Φ(x1)x1+

Φ(x∗1)

Φ(x1)

)+ αc2

(−x2x∗2

+Φ(x1)x1Φ(x∗1)x∗1

+ 1− x∗2x2

Φ(x1)x1Φ(x∗1)x∗1

)= c2r1

(1− x1

x∗1− Φ(x∗1)x∗1

Φ(x1)x1+

Φ(x∗1)

Φ(x1)

)+ αc2

(3− x2

x∗2− x∗2x2

Φ(x1)x1Φ(x∗1)x∗1

− Φ(x∗1)x∗1Φ(x1)x1

)+ αc2

(Φ(x∗1)x∗1Φ(x1)x1

+Φ(x1)x1Φ(x∗1)x∗1

− 2

)= S1 + S2 + S3.

Note that S2 ≤ 0, by AM −GM inequality, and that S2 = 0 if and only if x2 = x∗2and Φ(x1)x1 = Φ(x∗1)x∗1, which leads to x1 = x∗1. Let us now choose α = r1, so that

c2r1 = αc2. Then

S1 + S3 = c2r1

(−1− x1

x∗1+

Φ(x∗1)

Φ(x1)+

Φ(x1)x1Φ(x∗1)x∗1

)= c2r1

[x1x∗1

(Φ(x1)

Φ(x∗1)− 1

)− Φ(x∗1)

Φ(x1)

(Φ(x1)

Φ(x∗1)− 1

)]= c2r1

x1x∗1

(Φ(x1)

Φ(x∗1)− 1

)(1− Φ(x∗1)x∗1

Φ(x1)x1

)≤ 0,

since Φ is decreasing, while Ψ : (0,∞) → (0,∞), Ψ(x) = xΦ(x) = 1 − e−α1x

is increasing. By the previous analysis of the signs of S2 and T21, it follows that·U3 ≤ 0, with equality if and only if x1 = x∗1 and x2 = x∗2. The use of LaSalle’s

invariance principle finishes the proof of Theorem 5.1.

6. A caveat

Having in view the particular forms of (2.8), (2.9), (2.10) and (2.11) and, in fact, the

general structure of any model of commensalism, in which one equation is decoupled

from the other, one may erroneously rest assured that as the solution of the “simple”



equation converges to its steady state (monotonically, more so), the convergence of

the solution of the other equation is quite an easy exercise, perhaps under suitable

monotonicity assumptions. Why would then one need any other approach at all?

However, that is not always the case. Particularly, the convergence of the solution

of the “simple” equation does not inherently mitigate unboundedness, which always

lingers as a possible outcome in mutualistic and commensalistic systems. Below is

an example of a planar system that is related to Example 2.8 from Thieme [20],

slightly modified to produce a model that can be interpreted as a commensalistic

interaction between two species (to this purpose, note that the interaction term in

the right-hand side of the second equation of (6.1) is always positive, being null if

the size of the first species is equal to 0), but in which the limiting argument does

not work as expected.{x′ = −x+ 1,

y′ = −y(y − 1)(5− y) + yx(x− 1)2.(6.1)

A biological interpretation is that species y, the one which benefits from the

mutualistic interaction, has a feasible domain [0, 5]. In other words, we do not allow

an initial value for y greater than 5.

If one attempts to solve x first and then use its limit in the second equation, the

“conclusion” would be that (1, 1) is a globally stable equilibrium (i.e. global in the

feasible domain x > 0 and 0 < y < 5).

However this “conclusion” would not be true. The basin of attraction of (1, 1)

is not the same as far as y is concerned. In fact, it can be made arbitrarily small

the higher the initial value x0 is and y may even become arbitrarily large, as seen

below.

Solving for x we obtain from the second equation that

x(t) = (x0 − 1)e−t + 1.

Thus, the first equation can be viewed as a non-autonomous single ODE, in the

form

y′ ={−(y − 1)(5− y) + [(x0 − 1)e−t + 1](x0 − 1)2e−2t

}y.

Looking at the expression in the square brackets, we see that −(y−1)(5−y) < 0

for 1 < y < 5. However if x0 is chosen large enough, then initially, while t is still

close to zero, the negative component is canceled and y′ remains positive until y

increases past 5. Later on, as t increases, the effect of x0 is no longer important,

but since y(t) increased past 5, then y′ remains positive, and therefore y(t)→∞.

This calls for a higher degree of attention in the formalization of the limiting

procedure, and we shall achieve that in the next section, in the context of asymp-

totically autonomous systems.



7. The asymptotically autonomous systems approach

Apart from using Lyapunov’s second method, we may also use the theory of asymp-

totically autonomous systems to discuss the global stability of positive equilibria for

models of mutualism. To establish the framework for our approach, let us consider

the following differential systems

x′ = f(x, t) (7.1)

and

x′ = g(x), (7.2)

with t being the independent variable, t ∈ R, and x ∈ Rn. We say that (7.1) is

asymptotically autonomous with limiting system (7.2) if

f(x, t)→ g(x) as t→∞, locally uniformly with respect to x ∈ Rn.

The idea behind the use of asymptotically autonomous systems is often not to

remove the explicit dependence on t, since most models are already autonomous to

begin with, but rather to reduce the initial problem to a lower-dimensional one which

is significantly more tractable. However, extensive work done by Thieme and his

coworkers (see, for instance, Castillo-Chavez and Thieme [2] and references therein)

shows that the behavior of the limiting system may not always coincide with that

of the original one and specific conditions need to be imposed for this purpose.

In this regard, we provide below Theorems 2.3 and 2.5 from Castillo-Chavez and

Thieme [2], which characterize the connection between the original and the limiting

system in terms of the asymptotic behavior of orbits.

Theorem 7.1. Let e be a locally asymptotically stable equilibrium of (7.2) and

W its basin of attraction. Then every pre-compact orbit of (7.1) whose ω-limit set

intersects W converges to e.

Theorem 7.2. Let e1 and e2 two equilibria of (7.2). Assume that the space of

solutions (Rn) is the disjoint union of a closed set X1 and an open set X2 both

forward invariant under the flow of (7.1) and (7.2) such that e1 ∈ X1 and e2 ∈ X2.

Assume that e2 is locally stable for (7.2) and e1 is locally stable for the restriction

of (7.2) to X1. Assume also that e1 is a weak repeller under the flow of (7.1) (i.e.

no forward orbit of (7.1) starting in X2 converges to e1). Then every pre-compact

orbit of (7.1) starting in Xi converges to ei.

The important difference between these results is that only the second one (which

contains stronger assumptions) guarantees that the behavior of solutions of the

limiting system matches that of the original. Note that the first theorem does not

guarantee that the basins of attraction for e coincide between the two systems.

Counter-examples to that effect are provided in Castillo-Chavez and Thieme [2].

For our purpose, we take advantage of the fact that, in a model of commensal-

ism, the species that does not benefit from the other one can be analyzed separately



and have its asymptotic behavior characterized. Consequently, the equation mod-

elling the behavior of the second species can be considered as an asymptotically

autonomous differential equation, eligible for a treatment via the results mentioned

above.

7.1. A particular case

Let us consider the following two-dimensional models of commensalistic interactionsx′1 = r1x1

(1− x1

K1

),

x′2 = r2x2

(x2A2− 1

)(1− x2

K2

)+r2b21K2

x1x2

(7.3)

and x′1 = r1x1

(1− x1

K1

),

x′2 = r2x2

(x2A2− 1

)(1− x2

K2

)+ c2

(1− e−α1x1

)x2

(7.4)

that is, versions of (2.8) and (2.12), respectively, with strong Allee effects.

In what follows, we shall analyze them together by using the slightly more

general framework x′ = ax

(1− x

K

),

y′ = by( yA− 1)(

1− y

T

)+ f(x)y,

(7.5)

where f(x) is a continuous and positive valued function, a > 0, b > 0, K > 0 and

T > A > 0.

Since a > 0, it follows that x(t)→ K as t→∞ and the second equation becomes

an asymptotically autonomous ODE

y′ = g(t, y) := by( yA− 1)(

1− y

T

)+ f(x(t))y, (7.6)

its limiting ODE being given by

y′ = h(y) := by( yA− 1)(

1− y

T

)+ f(K)y. (7.7)

Since

|g(t, y)− h(y)| = |f(x(t))− f(K)||y|,

it is seen that g(t, y) converges locally uniformly to h(y).

The equilibria of (7.7) are 0 and any possible real root of the quadratic equation

− b

ATy2 +

(b

A+b

T

)y + f(K)− b = 0. (7.8)



Since the discriminant of (7.8) is

∆ =

(b

A− b

T

)2

+4b

ATf(K) > 0,

(7.8) has two real roots, which we may denote by y1 and y2, y1 < y2. We notice

that y1 + y2 > 0, and hence at least one of the roots is > 0.

Case 1. f(K) > b

In this case, y1 < 0 and y2 > 0. Furthermore, h(y) > 0 if 0 < y < y2 and h(y) < 0 if

y > y2, which implies that y2 is a globally stable equilibrium for the limiting ODE

(7.7).

We now prove that the solutions of (7.6) also converge to this equilibrium. First,

notice that [0,∞), the space of solutions of (7.7), is the disjoint union of the closed

set X1 = {0} and of the open set X2 = (0,∞), each of them containing the two

equilibria, e1 = 0 and e2 = y2 respectively. It is also easy to see that these two sets

are forward invariant under the flow of both (7.7) and (7.6).

Furthermore, 0 is a weak repeller for the asymptotically autonomous semi-flow

given by (7.6). This happens since if y(t)→ 0 with y(0) > 0, then

y′

y→ −b+ f(K) > 0,

and y(t) goes away from 0. Consequently, all conditions of Theorem 7.2 are met,

which means that all solutions of (7.6) converge to 0 or y2 and each of these equilibria

have the same basin of attraction as in (7.7). Since the basin of attraction of y2 is

the entire positive real line, it follows that y2 is a globally stable equilibrium point

for (7.6).

Remark 7.3. This case provides a threshold for the commensalism effect to over-

come the Allee effect of the species that benefits from the presence of the other.

Actually, when (x, y) approaches (K, 0), f(K) approximates the positive effects of

the mutualistic interaction upon the per capita growth rate of the second species,

while −b approximates the (negative) per capita growth rate in the absence of mu-

tualism. If f(K) > b, the positive effects dominate and the second species is able to

escape extinction.

To the contrary, the second case analyzed below, shows that, if this condition

fails, the Allee effect is preserved, even though the basin of attraction for the ex-

tinction steady state is still affected by the presence of commensalism.

Case 2. f(K) < b

In this case, both y1 and y2 are positive. Furthermore, since h(y) > 0 for y1 < y < y2and negative otherwise, it follows that for the limiting ODE (7.7) both 0 and y2 are

locally stable, while y1 is unstable. In other words, if 0 < y < y1, then y → 0 and



if y > y1, then y → y2. Hence the basin of attraction for 0 is [0, y1) and for y2 is

(y1,∞). Any solution of (7.6) starting at a point different than y1 will necessarily

intersect one of these two intervals. By Theorem 7.1, any solution of (7.7) starting

at a point different than y1 will then approach 0 or y2.

However, the conditions of Theorem 7.2 are not met (in particular, (y1,∞) is

not invariant under the flow of the asymptotically autonomous ODE).

7.2. The general case

We can further generalize the framework within (7.3) and (7.4) may be treated by

employing a generic function for the per-capita growth rate (i.e. without a separation

for the birth and the death rate), in the following form{x′1 = r1x1h1(x1),

x′2 = r2x2h2(x2, x1).. (7.9)

Let us consider the following assumptions upon the continuous functions h1 and h2

(C1) h1(x1) < 0 for all x1 > K1.

(C2) h2(0, 0) ≥ 0 and h2(x2, 0) < 0 for x2 > K2.

(C3) h2(x2, 0) < 0 for x < A2, h2(x2, 0) > 0 for A2 < x2 < K2 and h2(x2, 0) < 0 for

x > K2.

(C4) For any x1 > 0 there is Kx1> 0 such that h2(x2, x1) < 0 for all x2 > Kx1

.

(M) h2(x2, x1) is continuously differentiable and increasing in x1,

∂h2∂x1

(x2, x1) > 0.

Assumption (C1) ensures that all solutions of the first equation are bounded and

any given solution x1 converges to an equilibrium point, which is either 0 or a root of

h1(x1) = 0. Assumption (C2), formulated for the second species, is similar to (C1),

stating also that this species is not subject to a strong Allee effect. Assumption

(C3), dual to (C2), states that the second species is subject to a strong Allee effect

in the absence of commensalism.

Assumption (C4) ensures that the effects of commensalism do not lead to un-

bounded growth for the benefiting species, irrespective of the population size of

the other one. Finally, assumption (M) states that the model (7.9) describes indeed

a commensalism, since increasing the population of the first species has a benefic

effect upon the growth of the second one.

Let x∗1 be an equilibrium of the first equation and let x1 be a solution such that

x1(t) → x∗1 as t → ∞. Using the Mean Value Theorem and assumption (M), it

follows that h2 is h2 is locally Lipschitz as a function of x1 and consequently

|h2(x2, x1)− h2(x2, x∗1)| < B|x1 − x∗1|.

where B is a constant that depends only on the compact set containing x1 and

x2. Hence h2(x2, x1) converges locally uniformly to h2(x2, x∗1). From this, we can



consider x2 as described by an asymptotically autonomous differential equation

x′2 = r2x2h2(x2, x1), (7.10)

with limiting equation given by

x′2 = r2x2h2(x2, x∗1) (7.11)

Our main result is given as follows.

Theorem 7.4. Suppose (C1), (C2), (C4) and (M) hold for (7.9) and that for

each locally stable equilibrium x∗1 of the first equation of (7.9) there exist a unique

equilbrium x∗2 of (7.11). If x1(t)→ x∗1 then x2(t)→ x∗2 for any x2(0) > 0.

Before presenting the proof, note that we did not assume that x∗1 is unique

because this restriction is not necessary.

Proof. Notice that, by (M), h2(0, x∗1) > h2(0, 0) ≥ 0. Also, from (C4), h2(x2, x∗1)

is eventually negative. Hence, if x∗2 is unique, it is automatically a globally stable

equilibrium for the limiting equation (7.11).

We now verify the conditions of Theorem 7.2. Consider X1 = {0} and X2 =

(0,∞). Notice that X1 is closed, X2 is open and they form a disjoint partition of

[0,∞), the solution space of (7.11). Furthermore, 0 is a weak repeller for (7.10) since

if x2 → 0 for x2(0) > 0 then we would have

x′2x2

= r2h2(x2, x1) > r2h2(x2, 0)→ r2h2(0, 0) ≥ 0.

Therefore the basin of attraction of x∗2 in (7.10) and (7.11) is the same and

x2(t)→ x∗2 in (7.9).

If we assume a strong Allee effect for x2, that is, we replace assumption (C2)

with assumption (C3), we see that a necessary and sufficient condition to prevent

species x2 extinction is

h2(0, x∗1) > 0.

This ensures that 0 is always a repeller for x2. If this condition holds, the previous

theorem remains unchanged. However if h2(0, x∗1) < 0 then there will always be a

strong Allee threshold for x2, albeit one smaller than A2 due to the commensalistic

effect. In addition, there will always be at least two positive equilibria for x2 corre-

sponding to every x∗1. Then, from Theorem 7.1 we have that each solution of (7.10)

will converge to an equilibrium of (7.11) but, again, without the preservation of the

basins of attraction.



8. Conclusions

In this article, we analyzed from a stability viewpoint several models of two-species

commensalisms, which represent biological interactions in which only one species

benefits while the other is left unaffected. In particular, we chose models with

Richards and restricted growth rate, respectively, which are the commensalistic

equivalents of the mutualistic models analyzed in Georgescu et al. [7]. With sev-

eral modifications, made following an unitary procedure defined ad hoc, we showed

that the Lyapunov functionals employed in Georgescu and Zhang [6] and Georgescu

et al. [7] can be extended to prove global stability theorems of the corresponding

commensalistic models.

This extends the area of usefulness of these functionals in tackling stability

problems. This is particularly important for modelling problems since, on one hand,

global stability results give confidence to their biological interpretation but, on the

other hand, there is no universal mathematical algorithm to establish global stability

for all models enjoying this property. As an added benefit, these stability results,

within the context of commensalism, may be used to approximate the behavior of

two-species interaction in which one has only a negligible effect on the other.

Incorporating an Allee effect in the intrinsic growth rate of the positively in-

fluenced species, we have also investigated in the second part of the paper how a

commensalistic interaction prevents or maintains the possibility of population de-

cline or extinction. To this purpose, we have employed the theory of asymptotically

autonomous systems.

From a biological perspective, both above-mentioned outcomes can be impor-

tant. For example a commensalistic species can be introduced to prevent the ex-

tinction of an endangered species. In this regard, it has been observed in Halpern et

al. [10] that although positive interactions, in which one species benefits from the

presence of another species, such as mutualisms and commensalisms, are not usually

well integrated into contemporary approaches to aquatic restoration and conserva-

tion, they can often initiate recruitment and facilitation cascades which promote

an enhanced reproductive success. Further, positive interactions can be conducted

among populations and species across a wide range of scales (see Halpern et al.

[10], Table 1), since resources are often transferred between ecosystems via species

migration and the transport of organic nutrients. To this purpose, the spatial ar-

rangement, the connection between ecosystems, needs to be explicitly accounted for,

in order for the restoration of an ecosystem to be used to promote the enhancement

of others.

On the other hand, the interaction between humans and invasive species (acci-

dentally and/or intentionally introduced in a habitat) can be modeled by a com-

mensalistic interaction (human and invasive species) in which the invasive species

benefits from human actions. In such a case extinction may be desirable to prevent

the invasive species from establishing itself (Deacon et al. [4]).

A more comprehensive research that addresses these problems from an unifying



abstract viewpoint is currently underway and it will be reported in a a forthcoming

paper.

Acknowledgments

The work of P. Georgescu was supported by a grant of the Romanian National Au-

thority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-

2011-3-0557. D. Maxin acknowledges funding from Wheat Ridge Ministries – O. P.

Kretzmann Grant for Research in the Healing Arts and Sciences. The work of H.

Zhang was supported by the National Natural Science Foundation of China, Grant

ID 11201187, the Scientific Research Foundation for the Returned Overseas Chinese

Scholars and the China Scholarship Council.

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